The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first - and often inadequate - approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.
Mots-clés : Coarse-graining, a posteriori error estimate, relative entropy, lattice spin systems, Monte Carlo method, Gibbs measure, cluster expansion, renormalization group map
@article{M2AN_2007__41_3_627_0,
author = {Katsoulakis, Markos A. and Plech\'a\v{c}, Petr and Rey-Bellet, Luc and Tsagkarogiannis, Dimitrios K.},
title = {Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {627--660},
publisher = {EDP-Sciences},
volume = {41},
number = {3},
year = {2007},
doi = {10.1051/m2an:2007032},
mrnumber = {2355714},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007032/}
}
TY - JOUR AU - Katsoulakis, Markos A. AU - Plecháč, Petr AU - Rey-Bellet, Luc AU - Tsagkarogiannis, Dimitrios K. TI - Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 627 EP - 660 VL - 41 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007032/ DO - 10.1051/m2an:2007032 LA - en ID - M2AN_2007__41_3_627_0 ER -
%0 Journal Article %A Katsoulakis, Markos A. %A Plecháč, Petr %A Rey-Bellet, Luc %A Tsagkarogiannis, Dimitrios K. %T Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 627-660 %V 41 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007032/ %R 10.1051/m2an:2007032 %G en %F M2AN_2007__41_3_627_0
Katsoulakis, Markos A.; Plecháč, Petr; Rey-Bellet, Luc; Tsagkarogiannis, Dimitrios K. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 627-660. doi : 10.1051/m2an:2007032. http://www.numdam.org/articles/10.1051/m2an:2007032/
[1] , and, Renormalization group pathologies and the definition of Gibbs states. Comm. Math. Phys. 194 (1998) 359-388. | Zbl
[2] , Decay of correlations for infinite range interactions in unbounded spin systems. Comm. Math. Phys. 85 (1982) 517-528.
[3] , and, Spatially adaptive lattice coarse-grained Monte Carlo simulations for diffusion of interacting molecules. J. Chem. Phys. 121 (2004) 11420-11431.
[4] , and, Spatially adaptive grand canonical ensemble Monte Carlo simulations. Phys. Rev. E 71 (2005) 026702.
[5] and, Elements of Information Theory. John Wiley and Sons, Inc. (1991). | MR | Zbl
[6] and, Correlation functions of a lattice system. Comm. Math. Phys. 7 (1968) 274-288.
[7] , Lectures on Phase Transitions and the Renormalization Group, Volume 85. Addison-Wesley, New York (1992).
[8] and, General properties of polymer systems. Comm. Math. Phys. 22 (1971) 133-161.
[9] and, Mesoscopic modeling in the kinetic theory of adsorbates. J. Chem. Phys. 100 (1996) 19089.
[10] , and, Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamics properties. J. Chem. Phys. 118 (2003) 4414-4424.
[11] , and, Multiresolution analysis in statistical mechanics. II. Wavelet transform as a basis for Monte Carlo simulations on lattices. J. Chem. Phys. 118 (2003) 4424.
[12] , Scaling laws for Ising models near . Physics 2 (1966) 263.
[13] and, Information loss in coarse-graining of stochastic particle dynamics. J. Statist. Phys. 122 (2006) 115-135. | Zbl
[14] , and, Coarse-grained stochastic processes for microscopic lattice systems. Proc. Natl. Acad. Sci. 100 (2003) 782-782. | Zbl
[15] , and, Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems. J. Comp. Phys. 186 (2003) 250-278. | Zbl
[16] ,, and, Coarse-graining schemes for lattice systems with short and long range interactions. (In preparation).
[17] , and, Error analysis of coarse-graining for stochastic lattice dynamics. SIAM J. Numer. Anal. 44 (2006) 2270. | MR | Zbl
[18] and, Statistical Mechanics of Lattice Systems, Volume I. Springer Verlag (1999). | Zbl
[19] , Integral equations between distribution functions of molecules. J. Chem. Phys. 15 (1947) 187-201.
[20] , On Ising's model of ferromagnetism. Proc. Camb. Philos. Soc. 32 (1936) 477-481. | Zbl
[21] and, Coarse-graining limits in open and wall-bounded dissipative particle dynamics systems. J. Chem. Phys. 124 (2006) 184101.
[22] , and, A remark on high temperature polymer expansion for lattice systems with infinite range pair interactions. Lett. Math. Phys. 45 (1998) 303-322. | Zbl
[23] , The Statistical Mechanics of Lattice Gases, Vol. I. Princeton series in Physics (1993). | MR | Zbl
[24] , and, Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 (2001) 1169-1214. | Zbl
[25] , and, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Statist. Phys. 72 (1993) 879-1167. | Zbl
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